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It seems to me that some folks here are making the concept more complex than necessary.

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(My emphasis) Pretty good pun!

The reason for mathematical rigor about the real numbers goes back to Newton, who defined the derivative (what he called the fluxion) of a function by defining the difference quotient \(\frac{\Delta y}{\Delta x} \) then letting \(\Delta y\) and \(\Delta x\) get arbitrarily close to zero. He was able to use this idea to develop a theory of gravity that explained the falling of an apple to the earth as well as the motions of the planets in the sky. But he couldn't say what a fluxion really was. If \(\Delta y\) and \(\Delta x\) are nonzero, that's not the fluxion. But if they are both zero, then we get the nonsensical and undefined expression \(\frac{0}{0}\). Berkeley famously derided Newton's idea as the "ghosts of departed quantities."

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

Long (and interesting!) story short, it took mathematicians another 200 years to properly define what we today teach freshman calculus students as limits and derivatives. It's a shame IMO that we present students with the symbolic definitions but never tell them about the two century struggle to come up with those ideas.

One part of this struggle to put calculus on a rigorous logical footing was the proper definition of a real number. If you just say that the real numbers are "the numbers we use to do arithmetic" you get into all kinds of logical trouble when you try to define limits or deal with infinite sums. Because in fact limits and infinite sums go beyond arithmetic, and we need precise definitions in order to be sure we're reasoning correctly.

Another takeaway from this history is that physicists don't worry about making their math logically correct, as long as they get a good physical theory. Which is why one should not learn math from physicists.

I've explained several times that they're not the same thing. Decimal expressions are representations of reals, in the sense that the string "snow" is a representation of snow.

However, once we construct the reals (as Dedekind cuts or equivalence classes of Cauchy sequence or via any other such construction) we can then PROVE two things:

* Every real has a decimal expansion (and some have two); and

* Every decimal expansion corresponds to some real number (and sometimes two distinct decimal expressions correspond to the same real).

The proofs are standard and not very difficult. So we are justified in freely conflating reals and decimal expressions; although when we put on our philosopher hats, we realize that one is an abstract thing and the other is a representation.

No. Countable is a technical term in set theory. A set is countable if it can be placed in bijection with the natural numbers.

The word counted literally means that someone counted you. But in typically usage it has a political or social meaning. If you have a political opinion and you write letters to the editor or make speeches or contribute money to your favorite cause, you are "Standing up to be counted." That's what the world typically connotes in English. You're out there fighting the good fight. Another meaning is that you're reliable, you're someone who can be "counted on."

I wonder if this is a language issue. The distinctions you're making are not known to me,.

No.
No. Countable is a technical term in set theory. A set is countable if it can be placed in bijection with the natural numbers.
The word counted literally means that someone counted you. But in typically usage it has a political or social meaning. If you have a political opinion and you write letters to the editor or make speeches or contribute money to your favorite cause, you are "Standing up to be counted." That's what the world typically connotes in English. You're out there fighting the good fight. Another meaning is that you're reliable, you're someone who can be "counted on."
I wonder if this is a language issue. The distinctions you're making are not known to me,.

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LOL. Well, apparently, knowing a mathematical concept is not the same as understanding it.
All you need to understand what I said is to recognise the English word "counted" as the past particle of the verb to count. And know that a past participle can be used to signal a completed action.
And you don't understand that?!
Whoa, I'm impressed.
[/QUOTE]past participle
n. A verb form indicating past or completed action or time that is used as a verbal adjective in phrases such as finished work and baked beans and with auxiliaries to form the passive voice or perfect and pluperfect tenses in constructions such as The work was finished and She had baked the beans. Also called perfect participle.[/QUOTE]
You don't know this very basic piece of grammar?!

Makes no sense to me. A number or a function is computable if it is the output of some Turing machine. I have no idea what you mean by computed.
Are you going to clearly articulate the definitions you're using? Or just keep expressing your disbelief that not everyone knows what personal definitions you have in mind?

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Personal definition?? You must be kidding.
Look here:

countable (mathematics)
A term describing a set which is isomorphic to a subset of the natural numbers. A countable set has "countably many" elements. If the isomorphism is stated explicitly then the set is called "a counted set" or "an enumeration".

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counted (mathematics)
A term describing a set with an explicit isomorphism to the natural numbers.
Compare: countable.

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Freely available on the Internet...
I'm the French guy here but my English seems much better than yours.
Seems you have some work to do to catch up.

No.
No. Countable is a technical term in set theory. A set is countable if it can be placed in bijection with the natural numbers.

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"Counted" is also a technical term of Set Theory.
The question is whether the set is effectively "placed in bijection with the natural numbers".
All a proof that an infinite set is countable does is to prove that, or to give the principle of how, the set could be counted if there was an infinite amount of time to actually complete the counting.
If you don't understand that very basic distinction, I wonder what else you don't understand in all the stuff you're talking about.
EB

It might not be the formal mathematics definition, but the following is a reasonable description of the real numbers.

The positive real numbers correspond to distances along along an infinitely long line.

I addition to the above, zero & negative values of the above are real numbers.

They are the numbers a person ordinarily uses when doing arithmetic.​

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I don't believe people have any notion of the Reals. They probably all have the intuition of the continuum and presumably come to take the decimals, which they all have to learn, to be a good representation or approximation of the continuum. I don't think people go on about thinking whether the decimals is an exact representation of the continuum or not. Any notion of the Reals comes with mathematics and rather "late" in life. And most people just forget immediately about it because they don't need it. Anybody who really needs the Reals for his job probably has to pay close attention to the formal definition. In practice, I don't see that anyone but the mathematicians would need the Reals at all. I don't think that scientists need it except those who may want to see how you could go about proving the existence of continuums, probably not that many.
EB

Insults?!
I'm merely stating the obvious.
You said you did not understand what "counted" may mean in the context of mathematical sets, go on into irrelevancy by explaining I probably don't understand the connotations of the word and how it is used in English, and this even though "counted" is as much a mathematical concept as "countable" is. I really don't understand how it's possible for anyone who really understands the mathematical concept of countable, as indeed you've been suggesting in this thread, to fail to understand my mentioning of the word "counted" in the same context. It just beggars belief.
So, I don't believe you could possible explain yourself to anyone's satisfaction. You'd have to dissemble in some way. So, pretending to be offended by my post and quit is probably the least damaging solution available to you. Well done.
You will understand that all the mathematical views you've expressed at length here are rendered vacuous by your admission that you don't understand the mathematical concept of counted.
Still, I'll be a little bit more generous than that, at least with you personally. I think myself that the Reals are just beyond our ability to fully comprehend. Mathematicians are doing their best and it's clearly not enough and by a long shot. The problem, perhaps, it's that some of them, perhaps most, and you apparently, may be deluded that they understand the Reals properly. Well, that's human beings for you, nothing unusual here.
And I don't expect you could clarify, anyway, so it may be just as well to shut up.
EB

I believe that there are some erroneous views expressed in this Thread. The first line of SpeakPigeon Post 1 is essentially correct.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line?!

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It is not pedantically correct to use decimal numbers in the context of this discussion.

Octal, balanced trinary (base 3 using +, zero, & - as digits), or any other radix are just as also suitable although unfamiliar to many folks.​

It is more proper to discuss the issue in terms of the points on a an infinite length line without referring to a particular radix. Those points in addition to zero & the negative values of them correspond to the what are usually referred to as the real numbers.​

Sorry, James R. You are almost always correct in your Posts/Opinions. I do not remember considering any of your remarks/opinions being wrong. However, the following is incorrect (See remark below from Wolfram.com).

From your Post 2

That's a conjecture known as the continuum hypothesis.

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The above relates to SpeakPigeon remark

How could we even prove that things like the decimal numbers map any continuum?

The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers aleph_0 and the "large" infinite set of real numbers

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I believe that it is "un-decidable," but seems to be valid/true (& might be accepted as true by some knowledgeable folks).​

The following (also from your Post 2) is valid/true

There is, however, a rather ingenious proof that there is an uncountable infinity of real numbers (as compared to the countable infinity of the rationals, for instance).

From http://mathworld.wolfram.com/ ContinuumHypothesis I believe that it is "un-decidable," but seems to be valid/true (& might be accepted as true by some knowledgeable folks).​

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‎Gödel, who in 1940 proved CH consistent with the other axioms, thought CH was false. Cohen, who in 1963 proved that the negation of CH was consistent, also believed CH was false. Among those mathematicians who believe CH is even a meaningful question (some argue compellingly that it is not), most believe it's false.

CH by the way has nothing to do with whether the reals model the physical continuum. Alternatives to the standard reals for modeling the physical continuum are the constructive reals, the intuitionist reals, the hyperreals, perhaps the surreals. The philosopher Charles Sanders Peirce argued that no collection of set-theoretic points could possibly be a continuum. Perhaps there is no physical continuum at all, perhaps the world is discrete. But none of these philosophical issues have anything to do with CH, which is purely a question of abstract set theory.

Although Godel and Cohen may have believe CH was false, it still seems unprovable, since no one has come up with a cardinal between integer and real cardinals.

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Of course it's unprovable from the standard axioms. If one believes that CH is a meaningful question AND that it has a definite truth value in some Platonic universe of math (both philosophical questions) THEN most people who have thought about the issue believe it's false.

@Dinosaur made the claim that it's likely true, which is what I was responding to. Most mathematicians who have thought about the issue and who believe the question is both meaningful and that it has a definite truth value, believe it's false. A study of the literature will confirm what I say. The question of its provability is not at issue or under discussion here.

I'm sure you will agree that "Nobody has yet found X" is not a mathematical argument against X. Nobody's found a nontrivial zero of the zeta function off the critical line but that doesn't entitle this guy "Nobody" to the million dollar prize. Nobody's found the largest possible twin-prime pair but Nobody hasn't solved the twin prime conjecture either. This guy Nobody really gets around.

Cohen's heuristic argument against CH is interesting. He points out (in the concluding remarks to "Set Theory and the Continuum Hypothesis") that exponentiation is a very powerful operation, much more powerful than taking successors. For example between \(2^5\) and \(2^6\) there are many cardinals. [My example, not Cohen's]. Why shouldn't the same thing be true about transfinite exponentiation? Of course that's not a proof, but that was Cohen's published opinion. So when someone makes a claim that "CH must be true because such and so," they need to tell us what they know that ‎Gödel and Cohen didn't.

The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.

Most of these true/false/meaningless arguments are summarized or referenced in the Wiki article on CH, which many have linked but few seem to have actually read.